1. Note 3 Transmission Lines (Bounce Diagram)
ECE 3317
Prof. D. R. Wilton
Note Transmission Lines (Bounce Diagram)

2. Step Response
The concept of the bounce diagram is illustrated for a unit step response on a terminated line. RL
z = 0
z = L
V0 [V]
t = 0 + - Rg Z0 t

3. Step Response (cont.) The wave is shown approaching the load. t = 0Rg
t = 0 + RL
V0 [V] Z0 -
z = 0
z = L
(from voltage divider)

4. Bounce Diagram Rg t = 0 Z0 RL z + V0 [V] - z = L z = 0 T 2T 3T 4T 5T

5. Steady-State Solution
Adding all infinite number of bounces, we have:
Note: We have used
the geometric series
formula

6. Steady-State Solution
Simplifying, we have:

7. Steady-State Solution
Continuing with the simplification:
Hence we finally have:
Note: the steady-state solution does not depend on the transmission line length or characteristic impedance.
This is just the voltage divider equation!

8. Example Rg = 225 [] t = 0 RL = 25 [] Z0 = 75 [] T = 1 [ns] +
z = L
V0 = 4 [V]
t = 0 + -
Rg = 225 []
Z0 = 75 []
T = 1 [ns] 1 2 3 4 5 6

9. Example (cont.)
The bounce diagram can be used to get an “oscilloscope trace” at any point on the line.
0.75 [ns]
1.25 [ns]
2.75 [ns]
3.25 [ns]
steady state voltage:

10. Example (cont.)
The bounce diagram can also be used to get a “snapshot” of the line voltage at any point in time.
L/4

11. Example (cont.)
To obtain current bounce diagram from voltage diagram, multiply forward-traveling voltages by 1/Z0, backward-traveling voltages by -1/Z0.
voltage
current 1 2 3 4 5 6
Note: This diagram is for the normalized current, defined as Z0 I (z,t).

12. Example (cont.)
Note: We can also just change the signs of the reflection coefficients, as shown.
current
current 1 2 3 4 5 6 1 2 3 4 5 6
Note: These diagrams are for the normalized current, defined as Z0 I (z,t).

13. Example (cont.) current oscilloscope trace of current1 2 3 4 5 6
current
oscilloscope trace of current
0.75 [ns]
1.25 [ns]
2.75 [ns]
3.25 [ns]
steady state current:

14. Example (cont.) 1 2 3 4 5 6
current
L/4
snapshot of current

15. Reflection and Transmission Coefficient at Junction Between Two Lines
Example
Reflection and Transmission Coefficient at Junction Between Two Lines
junction
z = 0
RL = 50 []
z = L
V0 = 4 [V]
t = 0 + -
Rg = 225 []
Z0 = 75 []
Z0 = 150 []
T = 1 [ns]
(since voltage must be continuous across the junction)
KVL: TJ = 1 + J

16. Bounce Diagram for Cascaded Lines
Example (cont.)
Bounce Diagram for Cascaded Lines
Rg = 225 []
t = 0
T = 1 [ns]
T = 1 [ns] +
Z0 = 75 []
RL = 50 []
V0 = 4 [V]
Z0 = 150 [] -
z = 0
z = L 1 2 3
[V]
[V]
[V]
[V]
[V] 4

17. Pulse Response
Superposition can be used to get the response due to a pulse. RL
z = 0
z = L
Vg (t) + - Rg Z0 t W
We thus subtract two bounce diagrams, with the second one being a shifted version of the first one.

18. Example: Pulse Rg = 225 [] RL = 25 [] Z0 = 75 [] T = 1 [ns]
z = 0.75 L
z = 0
z = L
Rg = 225 []
Z0 = 75 []
T = 1 [ns]
Vg (t) + -
W = 0.25 [ns]
V0 = 4 [V] t W

19. Example: Pulse - W = 0.25 [ns] z = 0.75 L W 0.25 1 1.25 2 2.25 3 3.254 5 6
0.75 [ns]
1.25 [ns]
2.75 [ns]
3.25 [ns]
4.75 [ns]
5.25 [ns]
1.25
2.25
3.25
4.25
5.25
6.25 W
0.25
1.00 [ns]
1.50 [ns]
3.00[ns]
3.50[ns]
5.00 [ns]
5.50 [ns] -

20. Example: Pulse (cont.) Rg = 225 [] RL = 25 [] Z0 = 75 [] T = 1 [ns]
z = 0.75 L
z = 0
z = L
Rg = 225 []
Z0 = 75 []
T = 1 [ns]
Vg (t) + -
oscilloscope trace of voltage

21. Example: Pulse (cont.) - t = 1.5 [ns] W 1 2 3 4 5 6 1.25 2.25 3.25
4.25
5.25
6.25
0.25
L / 2 W
L / 4 -

22. Example: Pulse (cont.) t = 1.5 [ns] Rg = 225 [] RL = 25 []
z = 0
z = L
Rg = 225 []
Z0 = 75 []
T = 1 [ns]
Vg (t) + -
snapshot of voltage

23. Capacitive Load Z0 t = 0 C z = 0 z = L V0 [V] + -
Note: The generator is assumed to be matched to the transmission line for convenience (we wish to focus on the effects of the capacitive load).
Hence
The reflection coefficient is now a function of time.

24. Capacitive Load Z0
t = 0 +
V0 [V] Z0 CL -
z = L
z = 0 T 2T 3T t z

25. Capacitive Load (cont.)Z0
t = 0 +
V0 [V] Z0 CL -
z = L
z = 0
At t = 0: capacitor acts as a short circuit.
(valid for t < T)
At t = : capacitor acts as an open circuit.
Between t = 0 and t = , there is an exponential time-constant behavior.
Time-constant formula:
Hence we have:

26. Capacitive Load (cont.)Z0
t = 0 +
V0 [V] Z0 CL -
z = L
z = 0 t
V(0,t) T 2T
V0 / 2 V0
steady-state T 2T 3T t z

27. Inductive Load Z0 t = 0 LL At t = 0: inductor as a open circuit.
z = L
V0 [V]
t = 0 + - Z0
At t = 0: inductor as a open circuit.
(valid for t < T)
At t = : inductor acts as a short circuit.
Between t = 0 and t = , there is an exponential time-constant behavior.

28. Inductive Load (cont.) Z0 t = 0 LL z t V(0,t) t + V0 [V] - z = L z = 0
steady-state T 2T 3T t z

29. Time-Domain Reflectometer (TDR)
This is a device that is used to look at reflections on a line, to look for potential problems such as breaks on the line.
z = 0
load
z = L
V0 [V]
t = 0 + - Z0
(matched source) t
V (0, t) t
V (0, t)
resistive load, RL > Z0
resistive load, RL < Z0

30. Time-Domain Reflectometer (cont.)
z = 0
load
z = L
V0 [V]
t = 0 + - Z0
(matched source) t
V (0, t) t
V (0, t)
capacitive load
inductive load

31. Time-Domain Reflectometer (cont.)
Example of a commercial product
The 20/20 Step Time Domain Reflectometer (TDR) was designed to provide the clearest picture of coaxial or twisted pair cable lengths and to pin-point cable faults. 
AEA Technology, Inc.